Properties

 Label 2-234-117.16-c1-0-10 Degree $2$ Conductor $234$ Sign $-0.524 + 0.851i$ Analytic cond. $1.86849$ Root an. cond. $1.36693$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 − 2-s + (0.355 + 1.69i)3-s + 4-s + (−2.14 − 3.71i)5-s + (−0.355 − 1.69i)6-s + (−0.751 − 1.30i)7-s − 8-s + (−2.74 + 1.20i)9-s + (2.14 + 3.71i)10-s − 4.24·11-s + (0.355 + 1.69i)12-s + (−3.60 − 0.0419i)13-s + (0.751 + 1.30i)14-s + (5.53 − 4.95i)15-s + 16-s + (0.0242 − 0.0419i)17-s + ⋯
 L(s)  = 1 − 0.707·2-s + (0.205 + 0.978i)3-s + 0.5·4-s + (−0.959 − 1.66i)5-s + (−0.145 − 0.692i)6-s + (−0.283 − 0.491i)7-s − 0.353·8-s + (−0.915 + 0.401i)9-s + (0.678 + 1.17i)10-s − 1.27·11-s + (0.102 + 0.489i)12-s + (−0.999 − 0.0116i)13-s + (0.200 + 0.347i)14-s + (1.42 − 1.28i)15-s + 0.250·16-s + (0.00588 − 0.0101i)17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.524 + 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$234$$    =    $$2 \cdot 3^{2} \cdot 13$$ Sign: $-0.524 + 0.851i$ Analytic conductor: $$1.86849$$ Root analytic conductor: $$1.36693$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{234} (133, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 234,\ (\ :1/2),\ -0.524 + 0.851i)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$0.174530 - 0.312498i$$ $$L(\frac12)$$ $$\approx$$ $$0.174530 - 0.312498i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + T$$
3 $$1 + (-0.355 - 1.69i)T$$
13 $$1 + (3.60 + 0.0419i)T$$
good5 $$1 + (2.14 + 3.71i)T + (-2.5 + 4.33i)T^{2}$$
7 $$1 + (0.751 + 1.30i)T + (-3.5 + 6.06i)T^{2}$$
11 $$1 + 4.24T + 11T^{2}$$
17 $$1 + (-0.0242 + 0.0419i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (-2.87 + 4.98i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 + (-3.41 + 5.91i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 - 2.21T + 29T^{2}$$
31 $$1 + (-3.74 - 6.49i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 + (1.35 + 2.34i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + (3.68 - 6.39i)T + (-20.5 - 35.5i)T^{2}$$
43 $$1 + (2.89 + 5.00i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 + (1.67 - 2.89i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 - 1.82T + 53T^{2}$$
59 $$1 + 1.88T + 59T^{2}$$
61 $$1 + (0.877 + 1.51i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (-1.73 + 3.00i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 + (-0.645 + 1.11i)T + (-35.5 - 61.4i)T^{2}$$
73 $$1 + 12.9T + 73T^{2}$$
79 $$1 + (-4.24 + 7.35i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 + (0.370 - 0.642i)T + (-41.5 - 71.8i)T^{2}$$
89 $$1 + (-0.288 - 0.500i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 + (2.59 + 4.49i)T + (-48.5 + 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$